The Research Of Extreme Value Index And Properties Of Upper-Point In Extreme Value Theory

The properties of tails on distribution function is very significant, not only for extreme value theory, but also for financial and insurance theory. The extreme value index 7 of distribution function plays an important role in describing the properties of tails, and time distribution of financial time series mostly presets heavy-tailed. Therefore the estimation of tail index for heavy-tailed distribution have aroused our attention. Many scholars propose several methods, and get many different estimators. In chapter two of this paper, I will present a method based on Hill estimator, namely the situationγ< 0.In chapter three of paper, I will make an estimation for upper-point x~*, which can be seen in (γ|^)_n~-, and then get the relation between x~* and mean value of maximum statistics. In chapter four, I will provide the estimation for mean and variance of maximum statistics, and the relation between them and maximum statistics in the case of small sample, then prove it. At last by random simulation, I will demonstrate that the upper and lower bounds of maximum statistics are decided by their mean and variance. It's easily seen that for heavy-tailed distribution, the simulated upper bound of interval can well approach the upper-point x~*.